Often this is considered specifically for the case that the fiber sequence in Ch•(𝒜)Ch_\bullet(\mathcal{A}) is that induced from a short exact sequence in 𝒜\mathcal{A}. In this case the further map (that which makes the sequence “long”) is called the connecting homomorphism.

For if a chain mapA•→B•A_\bullet \to B_\bullet is the degreewise kernel of a chain map B•→C•B_\bullet \to C_\bullet, then if A^•→≃A•\hat A_\bullet \stackrel{\simeq}{\to} A_\bullet is a quasi-isomorphism (for instance a projective resolution of A•A_\bullet) then of course the composite chain map A^•→B•\hat A_\bullet \to B_\bullet is in general far from being the degreewise kernel of C•C_\bullet. Hence the notion of degreewise kernels of chain maps and hence that of short exact sequences is not meaningful in the homotopy theory of chain complexes in 𝒜\mathcal{A} (for instance: not in the derived category of 𝒜\mathcal{A}).

That short exact sequences of chain complexes nevertheless play an important role in homological algebra is due to what might be called a “technical coincidence”:

is not only a pushout square in Ch•(𝒜)Ch_\bullet(\mathcal{A}), exhibiting C•C_\bullet as the cofiber of A•→B•A_\bullet \to B_\bullet over 0∈C•0 \in C_\bullet, it is in fact also a homotopy pushout.

But a central difference between fibers/cofibers on the one hand and homotopy fibers/homotopy cofibers on the other is that while the (co)fiber of a (co)fiber is necessarily trivial, the homotopy (co)fiber of a homotopy (co)fiber is in general far from trivial: it is instead the loopingΩ(−)\Omega(-) or suspensionΣ(−)\Sigma(-) of the codomain/domain of the original morphism: by the pasting law for homotopy pullbacks the pasting composite of successive homotopy cofibers of a given morphism f:A•→B•f : A_\bullet \to B_\bullet looks like this:

cone(f)cone(f) is a specific representative of the homotopy cofiber of ff called the mapping cone of ff, whose construction comes with an explicit chain homotopyϕ\phi as indicated, hence cone(f)cone(f) is homology-equivalence to C•C_\bullet above, but is in general a “bigger” model of the homotopy cofiber;

And applying the chain homology functor to this yields the long exact sequence in chain homology which is traditionally said to be associated to the short exact sequence A•→B•→C•A_\bullet \to B_\bullet \to C_\bullet.

In conclusion this means that it is not really the passage to homology groups which “makes a short exact sequence become long”. It’s rather that passing to homology groups is a shadow of passing to chain complexes regarded up to quasi-isomorphism, and this is what makes every short exact sequence be realized as but a special presentation of a stage in a long homotopy fiber sequence.